We shall fix an instance of \(\mathbf {\omega }=(\mathbf {\omega }^a,\mathbf {\omega }^b)\)  w.r.t. \(\mathbb {P}_{a,b}\)  such that the cascade measures \(\bar{\mu }_{\mathbf {\omega }^a}\times \bar{\nu }_{\mathbf {\omega }^b}\)  and \(\bar{\theta }_{t,\mathbf {v}^-,\mathbf {\omega }}\)  for \(\theta \) -a.e. \((t,\mathbf {v}^-)\in \mathbb {T}\times \Sigma _{a,b}^-\)  are well-defined. The rest of the arguments are deterministic and we shall omit the notation \(\mathbf {\omega }\) .
Note that, since \(\theta \)  is \(T\) -invariant, \(\theta \)  and hence \(\theta ^-\)  projected to \(\mathbb {T}\)  is \(R\) -invariant. By unique ergodicity of irrational rotations, we get \(\theta ^-\)  projected to \(\mathbb {T}\)  is equal to the Lebesgue measure \(\ell \) , hence we may write
\(\theta ^-(\mathrm {d}(t,\mathbf {v}^-))=\theta ^-_t(\mathrm {d}\mathbf {v}^-)\ell (\mathrm {d} t),\) 
where we denote by \(\theta ^-_t\)  the disintegration of \(\theta ^-\)  w.r.t. \(\ell \) .
By Hunt and Kaloshin's analogue of Marstrand's projection theorem for measures ([1]}), we have that, for \(\ell \) -a.e. \(t\in \mathbb {T}\) , for \(\theta ^-_t\) -a.e. \(\mathbf {v}^-\in \Sigma _{a,b}^-\) , for the measure \(\bar{\theta }_{t,\mathbf {v}^-}\) , for Lebesgue almost every \(s\in \mathbb {R}\) ,
\(\underline{\dim }_H\pi _{s}S_t\Phi (\bar{\theta }_{t,\mathbf {v}^-})=\min \lbrace 1,\underline{\dim }_HS_t\Phi (\bar{\theta }_{t,\mathbf {v}^-})\rbrace .\) 
Together with Theorem REF  i) and Theorem REF , we obtain that for Lebesgue almost every \(s\in \mathbb {R}\) , \(E_\theta (\pi _s)=\min \lbrace 1,D_{\bar{\mu },\bar{\nu },\Phi }\rbrace \) .
On the other hand, by (REF ) we have for \(B\in \mathcal {B}(\mathbb {R})\)
\(\pi _s\Phi (\bar{\lambda })(B)=&\ \int _{\mathcal {M}(X)}\int _{\mathbb {T}}\int _{\Sigma _{a,b}^-} \pi _s\Phi (\bar{\theta }_{t,\mathbf {v}^-})(B)\,\theta ^-_t(\mathrm {d}\mathbf {v}^-)\ell (\mathrm {d}t)\Theta (\mathrm {d}\theta ).\) 

By (REF ) and Theorem REF  ii) this implies that
\(\underline{\dim }_H \pi _s\Phi (\bar{\mu }\times \bar{\nu }) \ge &\ \mathrm {ess inf}_{\theta \sim \Theta ,t\sim \ell ,\mathbf {v}^-\sim \theta ^-_t}\underline{\dim }_H \pi _s\Phi (\bar{\theta }_{t,\mathbf {v}^-})\\=&\ \mathrm {ess inf}_{\theta \sim \Theta ,t\sim \ell ,\mathbf {v}^-\sim \theta ^-_t}\underline{\dim }_H \pi _{s-t}S_t\Phi (\bar{\theta }_{t,\mathbf {v}^-})\\\ge &\ \mathrm {ess inf}_{\theta \sim \Theta , t\sim \ell } E_\theta (\pi _{s-t})\\=&\ \min \lbrace 1,D_{\bar{\mu },\bar{\nu },\Phi }\rbrace .\) 

But, by the exact-dimensionality of \(\Phi (\bar{\mu }\times \bar{\nu })\) , \(\min \lbrace 1,D_{\bar{\mu },\bar{\nu },\Phi }\rbrace \)  is also an upper bound for \(\overline{\dim }_B \pi _s\Phi (\bar{\mu }\times \bar{\nu })\) , hence the conclusion.
